Thick points for Gaussian free fields with different cut-offs

TL;DR

This paper investigates the fractal properties of thick points in Gaussian free fields with different cut-offs, establishing conditions for their Hausdorff dimension and comparing various approximation methods.

Contribution

It introduces new conditions for analyzing thick points of Gaussian free fields and compares different cut-offs to understand their fractal properties.

Findings

Established Hausdorff dimension formulas for thick points.

Proved that various cut-offs satisfy the necessary assumptions.

Provided criteria for comparing thick points across different cut-offs.

Abstract

Massive and massless Gaussian free fields can be described as generalized Gaussian processes indexed by an appropriate space of functions. In this article we study various approaches to approximate these fields and look at the fractal properties of the thick points of their cut-offs. Under some sufficient conditions for a centered Gaussian process with logarithmic variance we study the set of thick points and derive their Hausdorff dimension. We prove that various cut-offs for Gaussian free fields satisfy these assumptions. We also give sufficient conditions for comparing thick points of different cut-offs.